hw1 - Homework 1 Numerical Linear Algebra 1 Fall 2011...

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Homework 1 Numerical Linear Algebra 1 Fall 2011 Solutions will be posted Monday, 9/12/11 Problem 1.1 A matrix A C n × n is nilpotent if A k = 0 for some integer k > 0. Prove that the only eigenvalue of a nilpotent matrix is 0. Problem 1.2 Problem 7.1.1 Golub and Van Loan p. 318 Problem 1.3 Prove that a matrix A C n × n is normal if and only if there exists a unitary matrix U such that U H AU is a diagonal matrix. Problem 1.4 If A C n × n then the trace of A is the sum of the diagonal elements, i.e., trace( A ) = n i =1 e H i Ae i 1.4.a . Show that trace( AB ) = trace( BA ) where A C n × n and B C n × n . 1.4.b . Show that trace( Q H AQ ) = trace( A ) where A C n × n and Q C n × n is a unitary matrix. 1.4.c . Show that trace( A ) = n i =1 λ i where λ i are the eigenvalues of A . Problem 1.5 Recall that a nilpotent n × n matrix B is such that B k = 0 for some k n . Suppose A C n × n (not necessarily Hermitian) and show that
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